Boundary value problems for elliptic differential-difference equations have some astonishing properties. For example, unlike elliptic differential equations, the smoothness of the generalized solutions can be broken in a bounded domain and is preserved only in some subdomains. The symbol of a self-adjoint semibounded functional differential operator can change its sign. The purpose of this book is to present for the first time general results concerning solvability and spectrum of these problems, a priori estimates and smoothness of solutions. The approach is based on the properties of elliptic operators and difference operators in Sobolev spaces. The most important features distinguishing this work are applications to different fields of science. The methods in this book are used to obtain new results regarding the solvability of nonlocal elliptic boundary value problems and the existence of Feller semigroups for multidimensional diffusion processes. Moreover, applications to control theory and aircraft and rocket technology are given. The theory is illustrated with numerous figures and examples. The book is addresssed to graduate students and researchers in partial differential equations and functional differential equations. It will also be of use to engineers in control theory and elasticity theory.

Inhalt

I Boundary Value Problems for Functional Differential Equations in One Dimension.- 1 Ordinary Differential Equations with Nonlocal Boundary Conditions.- 2 Difference Operators in One Dimension.- 3 The Boundary Value Problem for the Differential-Difference Equation.- 4 Generalized and Classical Solutions.- 5 Applications to Control Systems with Delay.- 6 The Boundary Value Problem for the Differential-Difference Equation with Degeneration.- Notes.- II The First Boundary Value Problem for Strongly Elliptic Differential-Difference Equations.- 7 Some Geometrical Constructions.- 8 Difference Operators in the Multidimensional Case.- 9 Necessary and Sufficient Conditions for Strong Ellipticity.- 10 Solvability and Spectrum.- 11 Smoothness of Generalized Solutions in Subdomains.- 12 Smoothness of Solutions on a Boundary of Neighboring Subdomains.- 13 Elliptic Differential Equations with Nonlocal Conditions on Shifts of Boundary.- Notes.- III Applications to the Mechanics of a Deformable Body.- 14 The Elastic Model.- 15 Variational and Boundary Value Problems.- 16 Smoothness of Solutions.- 17 The One-Dimensional Case.- Notes.- IV Semi-Bounded Differential-Difference Operators with Degeneration.- 18 Self-Adjoint Extension of a Semi-Bounded Differential-Difference Operator.- 19 The Spectrum of Semi-Bounded Differential-Difference Operators.- 20 Smoothness of Solutions of Equations with Degeneration.- Notes.- V Nonlocal Elliptic Boundary Value Problems.- 21 Nonlocal Elliptic Problems with a Parameter.- 22 Elliptic Equations with Nonlocal Boundary Conditions in a Cylinder.- 23 Elliptic Differential-Difference Equations in a Cylinder.- 24 Applications to the Multidimensional Diffusion Processes.- 25 Elliptic Problems with Nonlocal Conditions near the Boundary and Feller Semigroups.- Notes.- A Linear Operators.- B Functional spaces.- C Elliptic Problems.- List of Symbols.